Download Linear inverse problems on Erd˝os

Linear inverse problems on Erdős-Rényi graphs:
Information-theoretic limits and efficient recovery
Emmanuel Abbe, Afonso S. Bandeira, Annina Bracher, Amit Singer
Princeton University
Abstract—This paper considers the linear inverse problem
Y = AX ⊕ Z, where A is the incidence matrix of an
Erdős-Rényi graph, Z is an i.i.d. noise vector, and X is the
vector of unknown variables, assumed to be Boolean. This
model is motivated by coding, synchronization, and community
detection problems. Without noise, exact recovery is possible
if and only the graph is connected, with a sharp threshold
at the edge probability log(n)/n. The goal of this paper is
to determine how the edge probability p needs to scale in
order to cope with the noise. Defining the rate parameter
r = log(n)/np, it is shown that for an error probability of
ε close to half, exact recovery is possible if and only if r
is below D(1/2||ε). In other words, D(1/2||ε) provides the
information theoretic threshold for exact recovery at lowsnr. In addition, an efficient recovery algorithm based on
semidefinite programming is proposed and shown to succeed
up to half the threshold.
I. I NTRODUCTION
A large variety of problems in information theory, machine learning, and image processing are concerned with
inverse problems on graphs, i.e., problems where a graphical
structure governs the dependencies between the variables
that are observed and the variables that are unknown. In
simple cases, the dependency model is captured by an
undirected graph with the unknown variables attached at the
vertices and the observed variables attached at the edges. Let
G = (V, E) be a graph with vertex set V and edge set E,
and let xV be the vertex- and y E the edge-variables. In many
cases of interest (detailed below), the probabilistic model for
the edge-variables conditionally on the vertex-variables has
a simple structure: it factorizes as
Y
P (y E |xV ) =
Q(ye |x[e]),
(1)
e∈E
where ye denotes the variable attached to edge e, x[e]
denotes the two vertex-variables incident to edge e, and
Q is a local probability kernel. In this paper, we consider
Boolean edge- and vertex-variables, and we assume that the
kernel Q is symmetric and depends only on the XOR of the
vertex-variables.1 The edge-variables can then be viewed as
a random vector Y E that satisfies
Y E = BG xV ⊕ Z E ,
(2)
where BG is the incidence matrix of the graph, i.e., the
|E| × |V | matrix with (BG )e,v = 1 if and only if vertex v
1 Symmetry means that Q(y|x , x ) = P (y|x ⊕ x ) for some P that
1
2
1
2
satisfies P (1|1) = P (0|0).
belongs to edge e, and Z is a random vector of dimension
|E| representing the noise.
In the above setting, the forward problem of recovering
the most likely edge-variables given the vertex-variables is
trivial and amounts to maximizing Q for each edge. The
inverse problem, however, is more challenging: the most
likely vertex-variables (say with a uniform prior) given the
edge-variables cannot be found by local maximization. This
type of problem arises in various contexts:
Coding: For a (symmetric) kernel Q, equation (2) corresponds to the output of a simple code, namely a 2-rightdegree LDGM code, over a binary (symmetric) channel.
While this is not a particularly interesting code by itself
(e.g., at any fixed rate, it has a constant fraction of isolated
vertices), it is a relevant primitive for the construction of
other codes such as LT or raptor codes [18], [19].
Constraint satisfaction problems: The model (1) is a
particular case of the graphical channel studied in [2] in
the context of hypergraphs. This class of models allows in
particular to recover instances of planted constraint satisfaction problems (CSPs) by choosing uniform kernels Q,
where the vertex-variables represent the planted assignment
and the edge-variables represent the clauses. In the case of
a simple graph and not a hypergraph, this provides a model
for planted formulae like 2-SAT or 2-XORSAT (model (2)).
Synchronization: Equation (2) results also from the
synchronization problem studied in [20], [6], [24], [3], [7],
if the dimension is one, i.e., each vertex-variable is the 1bit quantization of the reflection of a signal. The goal in
synchronization over O(d), the group of orthogonal matrices
of size d × d, is to recover exactly the original values of
the node-variables {xj }j∈[n] , assumed to take values in
O(d), given the relative measurements {Jij x−1
i xj }i,j∈[n] ,
where Jij is randomly drawn in O(d) if the vertices i
and j are adjacent and all-zero else.2 When d = 1, we
have O(1) = {−1, +1} and the synchronization problem is
equivalent to (2).
Community detection: The model in (1) can also be
used as a probabilistic network model. The basic ErdősRényi model [11], is typically not a good model for networks: all vertices have the same expected degree and no
cluster structure appears. One possibility to obtain cluster
structures is to attach latent variables to the vertices and
assume an edge distribution that depends on these variables.
2 If
Jij is the d × d identity matrix, then the measurement is noise-free.
There are various models with latent variables, such as the
exchangeable, inhomogeneous or stochastic block models
[12], [25], [13], [10], [15], [9]. The model in (1) can be
used for this purpose. The vertex-variables represent the
community assignment, the edge-variables the connectivity,
and the graph G the information is available. In the noiseless
additive case and for every edge e = (i, j), the edgevariable ye = xi ⊕ xj encodes whether the vertices i and
j are in the same community or not. With noise on top,
vertices in the same/different communities are also allowed
disconnect/connect, which is a more realistic model.
In this paper we are interested in particular in a “filtered”
block model, as introduced in [2] in a different context. In
such a model, a base-graph G = (V, E(G)) and a binary
community assignment X ∈ {0, 1}V are used to generate a
random graph on the vertex set V with ternary edge labels
Eij ∈ {∗, 0, 1} drawn independently with the following
probability distribution:
P{Eij = ∗|E(G)ij = 0} = 1
(3a)
P{Eij = 1|Xi = Xj , E(G)ij = 1} = q1 ,
(3b)
P{Eij = 1|Xi 6= Xj , E(G)ij = 1} = q2 .
(3c)
Put differently, (3) is a graph model where information is
only available on the base-graph G, the ∗-variable encodes
the absence of information, and when there is information
available, then two vertices are connected with probability
q1 if they are in the same community and with probability
q2 if they are in different communities. When G = Kn is
the complete graph and X is uniformly distributed, this is
the standard stochastic block model with two communities
[8], [17]. In the case of (2), the linear structure implies that
q1 = 1 − q2 = ε.
While all the above mentioned problems are concerned
with related inverse problems on graphs, there are various
refinements that can be considered for the recovery. This paper focuses on exact recovery, which requires that all vertexvariables be recovered simultaneously with high probability
as the number of vertices diverges. The probability measure
may depend on the graph ensemble or simply on the kernel
Q if the graph is deterministic. Note, however, that exact
recovery of all variables in the model (2) is not quite
possible: the vertex-variables xV and 1V ⊕ xV produce
the same output Y E . Exact recovery is meant “up to a
global flipping of the variables”. For partial recovery, only a
constant fraction of the vertex-variables are to be recovered
correctly with high probability as the number of vertices
diverges. Put differently, the true assignment need only be
positively correlated with the reconstruction. The recovery
requirements vary with the applications, e.g., exact recovery
is typically required in coding theory to ensure reliable
communication. In community detection, partial recovery
is the best one can hope for sparse networks, while exact
recovery can be considered for slightly denser graphs with
logarithmic degree [16].
This paper focuses on exact recovery for the linear
model (2) with Boolean variables, and on Erdős-Rényi
models for the base-graph G. For this setup, we identify
the information-theoretic phase transition for exact recovery
in terms of the edge density of the graph and the noise
level, and devise an efficient algorithm based on semidefinite
programming (SDP), which approaches the threshold up to
a factor of 2. This SDP based method was first proposed in
[20], and it shares many aspects with the SDP methods in
several other problems [21], [14].
II. M ODEL AND RESULTS
In this paper, we focus on the linear Boolean model
Y E = BG xV ⊕ Z E ,
(4)
where the vector components are in {0, 1} and the addition
is modulo 2. We require exact recovery for xV arbitrary and
consider for the underlying graph G = (V, E) the ErdősRényi model ER(n, p), where V = [n] and the edges are
drawn i.i.d. with probability p. We assume that the noise
vector Z E has i.i.d. components, equal to 1 with probability
ε. We assume3 w.l.o.g. that ε ∈ [0, 1/2], where 0 means no
noise (and exact recovery amounts to having a connected
graph) and 1/2 means maximal noise (and exact recovery is
impossible no matter how connected the graph is). Note that
the inverse problem is much easier if the noise model causes
erasures with probability ε instead of errors. Exact recovery
is then possible if and only if the graph is still connected
after the erasure of the edges. Since there is a sharp threshold
for connectedness at p = log(n)
n , this happens a.a.s. if p =
(1+δ) log(n)
for some δ > 0. Hence 1−ε is a sharp threshold
n(1−ε)
in log(n)/np for the exact recovery problem with erasures
and base-graph ER(n, p).
The goal of this paper is to find the replacement to the
erasure threshold 1−ε for the setting where the noise causes
errors. Similarly to channel coding where the Shannon
capacity of the BSC(ε) differs from the BEC(ε) capacity,
we obtain for the considered inverse problem the expression
D(1/2||ε) for the low-snr regime (ε close to 1/2).
More precisely, this paper establishes an information
theoretic (IT) necessary condition that holds for every
graph (Theorem IV.1), and an IT (Theorem IV.2) sufficient
condition for Erdős-Rényi graphs. Moreover, we also give
a recovery guarantee that holds for an efficient algorithm
based on SDP (Theorem V.3).
In particular, we show that, for ε = 1/2 − ∆n , where
∆n = o(1) and ∆n = Ω(n−τ ) for every τ > 04 , it is
optimal to draw the graph from the Erdős-Rényi model: The
bounds for the necessary condition for a general graph and
the IT sufficient condition for the Erdős-Rényi graph match,
and they match the SDP bound up to a factor 1/2.
If the noise parameter ε is bounded away from both zero
and 1/2, then all conditions imply m/n = Θ(log(n)), where
3 The noise model is assumed to be known, hence the regime ε ∈ [1/2, 1]
can be handled by adding an all-one vector to Y E .
4 The assumption ∆ = o(1) is standard for the synchronization problem
n
in dimension d = 1.
m is the expected number of edges: m = n2 p. The factors
by which the bounds differ decrease with an increasing noise
parameter ε. Since in the noise-free case exact recovery is
possible if and only if the graph is connected, which is
true for trees with m = n − 1 and for Erdős-Rényi graphs
with m = n log(n) /2, the factors between the necessary
condition and the sufficient conditions approaches infinity
when ε decreases to zero (since D(1/2||ε) diverges).
III. D IRECTIONS AND OPEN PROBLEMS
There are various extensions to consider for the above
models, including the generalization to q-ary instead of
binary variables and the extension to problems with hyperedges instead of edges as in [2]. Non-binary variables would
be particularly interesting for the synchronization problem
in higher dimension, d ≥ 2, where the orientations are
quantized to a higher order. There are several extensions
that are interesting for the community detection applications.
First, it would be important to investigate non-symmetric
noise models, i.e., noise models that are non-additive. Then,
it would be interesting to study partial (as opposed to
exact) recovery for sparse graphs with constant degrees, and
to incorporate constraints on the size of the communities.
Finally, one could consider other ensembles for the basegraph. In the full version of this paper, which will appear
in [1], the spectral gap of the base-graph is used to analyze
the threshold behaviour for any deterministic graph.
IV. I NFORMATION T HEORETIC B OUNDS
This section presents necessary and sufficient conditions
for exact recovery of the vertex-variables xV from the edgevariables Y E . We require that ML-decoding recover the
V
vertex-variables
V V x up to an unavoidable additive offset
φ ∈ 0 ,1
with some probability that converges to 1
as the number of vertices approaches infinity.
Notation: All logarithms have base e, and we denote
by D(1/2 ||ε ) = 1/2 log(1/(2ε)) + 1/2 log(1/(2(1 − ε)))
the Kullback-Leibler divergence between 1/2 and ε and by
hb (q) = q log(1/q) + (1 − q) log(1/(1 − q)) the entropy (in
nats) of a binary random variable that assumes the value 1
with probability q ∈ [0, 1].
A. A Necessary Condition for Successful Recovery
For each graph G = (V, E) (drawn from the Erdős-Rényi
model or not), the following result holds:
Theorem IV.1. Necessary conditions for exact recovery are:
Let 0 < τ < 2/3. If m/n ≤ nτ , then
1 − 3τ /2
log n
m
≥
log(n) + o
. (5)
n
2D(1/2 ||ε )
D(1/2 ||ε )
If ε → 1/2, this implies for (1 − 2ε) ≥ n−τ /2
m
1 − 3τ /2
log n
≥
log(n) + o
2
2
n
(1 − 2ε)
(1 − 2ε)
incident to Vertex vj are noisy. As we argue next, if event
Ej occurs, then ML-decoding recovers vertex-variables other
than xV or xV ⊕ 1V with probability at least 1/2. Indeed,
if ML-decoding correctly recovers the vertex-variables that
are attached to the vertices adjacent to vj up to a global
additive offset φ ∈ {0, 1}, then—by assumption that event
Ej occurs—the probability that ML-decoding recovers xj
with offset φ⊕1 is at least 1/2. In particular, this implies
that
T
ML-decoding can only be successful if the event vj ∈V Ejc
occurs. Let Q be a subset of [n] such that no two vertices
with indices in Q are adjacent. Since the noise Z E is
drawn IID, the events {EjT}j∈Q are independent and the
c
probability of the event
j∈Q Ej is easily computable.
T
T
c
Moreover, the event j∈[n] Ej can only occur if j∈Q Ejc
occurs. A necessary condition for
T exact recovery thus is
that the probability of the event j∈Q Ejc converges to one
as the number of vertices increases. In the following, we
prove the claim by identifying
T a set Q and upper-bounding
the probability of the event j∈Q Ejc .
Let deg(vj ) be the degree of Vertex vj , and assume w.l.g.
deg(v1 ) ≤ deg(v2 ) ≤ . . . ≤ deg(vn ). For every 0 < δ ≤ 1
n
X
2m ≥
.
(6)
Proof: Fix a vertex vj , and let Ej denote the event that
the variables attached to at least half of the edges that are
(7)
j=dδne
For j ≤ dδne, we therefore find
2m
2m
≤
.
deg(vj ) ≤ deg vdδne ≤
d(1 − δ) ne
(1 − δ) n
(8)
This implies that for every set L ⊆ {1, . . . , dδne}, the
vertices {vj : j ∈ L} are disconnected from at least
2m
dδne − |L| 1 +
(9)
(1 − δ) n
vertices in the
set {vj : j ≤ dδne}. In particular, there is
a set Q ⊆ dδne such that no two distinct vertices with
indices in the set Q are adjacent and such that
|Q| ≥
dδne
δ (1 − δ) n2
≥
.
2m
2m + (1 − δ) n
1 + (1−δ)n
(10)
If j ≤ dδne, then one can show that
deg(vj )
X
deg(vj ) k
deg(vj )−k
Pr[Ej ] =
ε (1 − ε)
k
deg(vj )
2
k=
1−ε
2mD(1/2||ε )
≥ e− 2 log(( ε ) (1−δ)n )−log(2)− (1−δ)n . (11)
Since the events Ejc : j ∈ Q are jointly independent,


Y
\
Pr
Ejc  =
(1 − Pr[Ej ])
1
j∈Q
!
deg(vj ) ≥ d(1 − δ) ne deg vdδne .
a)
≤ e−
b)
≤e
2m
j∈Q
P
j∈Q
log
−e
e
2mD(1/2||ε )
2m
−log(2)−
(( 1−ε
ε ) (1−δ)n )
(1−δ)n
− 1 log
2
δ(1−δ)n2
4m+2(1−δ)n
√ (1−δ)n √
2m
!
ε
1−ε
−
2mD(1/2||ε )
(1−δ)n
,
(12)
where a) holds since 1 − x ≤ e−x for x ≥ 0 and because of
−1
(11), and b) is due to (10). To conclude, take δ = log(n) ,
τ
and observe that for m/n ≤ n , where τ < 2/3, the RHS
of (12) is bounded away from 1 unless (5) holds.
It is interesting to compare Theorem IV.1 to the nec1
essary condition m
n ≥ 1−hb(ε)/ log 2 , previously shown in
[20, Section 5]. If ε ∈ (0, 1/2) does not depend on n,
then this condition only implies m/n = Ω(1) and is
thus weaker than m/n = Ω(log n), which follows from
Theorem IV.1. If ε = 1/2 − ∆n and ∆n = o(1), then
hb (ε) = 1 − 2∆2n / log 2 + op∆2n , and we can write the
condition in [20] as ∆n = Ω( n/m). If there is a τ < 2/3
−τ /2
such that ∆n ≥ n
p , then Theorem IV.1 is tighter: it
implies ∆n = Ω( log(n) n/m). However, if there is no
such τ , then Theorem IV.1 cannot be applied.
B. A Sufficient Condition for Successful Recovery
We next present a sufficient condition for exact recovery.
For a random base-graph G = (V, E) from the Erdős-Rényi
model, we require the vertex-variables xV to be recoverable
from the edge-variables Y E except with some probability
that vanishes as the number of vertices increases.
Notation: Recall that G is the underlying graph on n
nodes, and let H be the subgraph representing the incorrect
edges (corresponding to Z(i,j) = 1). Let AG , AH , DG ,
DH , LG , and LH be, respectively, the adjencacy, degree,
and Laplacian matrices of the graphs G and H.
As in [20], we assume w.l.o.g. that xV ≡ 0 so that g ≡
1. Then, AG − 2AH is the matrix whose (i, j)-th entry is
equal to ρij if (i, j) ∈ E and 0 otherwise.
Problem (15) has
the same solutions as maxgi ∈{±1} Tr (AG − 2AH )gg T ,
which in turn is equivalent to
max Tr [(AG − 2AH )X]
(16)
s.t. X ∈ Rn×n , Xii = 1 ∀i, X 0, Rank(X) = 1.
(Given the optimal rank 1 solution X of (16), gi = (−1)xi is
the only non-trivial eigenvector of X.) As the rank constraint
is non-convex, we consider the following convex relaxation
max Tr [(AG − 2AH )X]
s.t. Xii = 1, X 0.
(17)
Note that (17) is an SDP and can be solved, up to arbitrary
precision, in polynomial time [23]. Note that a solution of
(17) need not be rank 1 and thus need not be a solution of
(16). However, we will show that under certain conditions
Theorem IV.2. If the base-graph is from the Erdős-Rényi (17) recovers the same optimal solution as (16). In this case,
model ER(n, p) with p > 2 log(n)/n, then the condition
gi = (−1)xi is the only non-trivial eigenvector of X and
xV can be recovered via the tractable problem (17).
1
log(n)
m
q
≥ log(n)+o
Our objective is to understand when X = gg T = 11T is
log(n)
n
D(1/2
||ε
)
2 1−
D(1/2 ||ε )
the unique optimal solution to (17). The dual of the SDP is
m/n
(13)
min Tr(Z) s.t. Z diagonal, Z − (AG − 2AH ) 0. (18)
is sufficient for exact recovery. If → 1/2 the condition is
!
Duality guarantees that the objective value of (17) cannot
m
1
log(n)
exceed that of (18). Thus, if there exists Z, feasible solution
≥
log(n)
+
o
.
(14)
2
2
n
(1 − 2)
(1 − 2ε)
of (18), such that Tr(Z) = Tr (AG − 2AH ) 11T , then
X = 11T is an optimal solution of (17). Moreover, Z and
Proof: One can prove the theorem using
ML V that
V
V
V
11T have to satisfy complementary slackness: Tr(11T (Z −
decoding succeeds if every
tuple x̃ ∈
/ x , x ⊕ 1
E
V
E
V
(AG − 2AH ))) = 0. Given these constraints, a natural
satisfies dH Y , BG x̃ > dH Y , BG x .
candidate is Z = DG − 2DH . Indeed, it is easy to see
V. C OMPUTATIONALLY EFFICIENT RECOVERY - THE SDP that Tr(DG − 2DH ) = Tr (AG − 2AH ) 11T . Hence, if
In this section we analyze a tractable method to
LG − 2LH = DG − 2DH − (AG − 2AH ) 0,
(19)
recover xV from the noisy measurements Y E based
in semidefinite programming. Ideally, one would like i.e., the dual variable satisfies the PSD constraint), then 11T
to find the maximum
likelihood estimator x∗ = must be an optimal solution of (17). Additionally, if LG −
P
argminxi ∈{0,1} (i,j)∈E 1{xi 6=y(i,j) ⊕xj } . By defining the 2LH is not only PSD but also its second smallest eigenvalue
{±1}-valued variables gi = (−1)xi and the coefficients is non-zero, then it is not difficult to show that 11T is the
unique optimal solution. We have thus shown:
ρij = (−1)y(i,j) , the ML problem is reformulated as
X
min
|gi − ρij gj |2 .
(15) Lemma V.1. If
gi ∈{±1}
(i,j)∈E
This problem is known to be NP-hard in general (in fact, it
is easy to see that it can encode Max-Cut). In what follows,
we will describe and analyze a tractable algorithm, which
was first proposed in [20] to approximate the solution of
(15). We will state conditions under which the algorithm
is able to recover the vertex-variables xV . The idea is to
consider a natural semidefinite relaxation. Other properties
of this SDP have been studied in [4], [5].
LG − 2LH 0 and λ2 (LG − 2LH ) > 0,
(20)
then 11T is the unique solution to (17).
Remark V.2. A similar exact recovery sufficient condition
was independently obtained by Huang and Guibas [14] in
the context of consistent shape map estimation (see Theorem
5.1. in [14]). Their analysis goes on to show, essentially, that
as long as the probability of a wrong edge is strictly smaller
than 12 , the probability of exact recovery converges to 1 as
the size of the graph is arbitrarily large. On the other hand,
we are able to show near tight rates at which this phase
transition happens. For a given that is arbitrarily close to
1
2 we give an essentially-tight bound on the size of the graph
and edge density needed for exact recovery (Theorem V.3).
A. Erdős-Rényi Model
We now assume that the underlying graph is drawn from
the Erdős-Rényi model ER(n, p) and use condition (20) to
give guarantees for exact recovery.
For each pair of vertices i < j, let Λij be the matrix
that is 1 in the entries (i, i) and (j, j), −1 in the entries
(i, j) and
P (j, i), and 0 elsewhere. Observe that Λij 0, and
LG = i<j:(i,j)∈E Λe . Let αij be the random variable that
takes the value 0 if Edge (i, j) is not in G, the value 1 if it
is in G but not in H, and the value −1 if it is in H. Hence
αij are iid and take the values 0, 1, −1 with probability,
respectively, 1 − p, p (1 − ε), and pε.
P
In the new notation, LG − 2LH =
i<j αij Λij .
We define the centered random
variables
Aij =
P
(p(1 − 2ε) − αij ) Λij . For A =
A
,
we
can
write
i<j ij
LG − 2LH = p(1 − 2ε)(nI − 11T ) − A. Since Λij always
contains the vector 1 in the null-space, (20) is equivalent to
λmax (A) < p(1 − 2ε)n.
We next argue for which values p, ε, and n there is some
δ > 0 such that Prob [λmax (A) ≥ p(1 − 2ε)n] ≤ n−δ .
To this end, we use the Matrix Bernstein inequality (Theorem 1.4 in [22]),
Prob [λmax(A) ≥ t] ≤
which impliesP
2
/2
, where σ 2 = i<j EA2ij , with k · k
n exp − σ2t+Rt/3
denoting the
spectral norm,
and R ≥ λmax (Ak ). Note that
σ 2 = 2np 1 − p(1 − 2ε)2 and λmax (Aij ) ≤ 2p(1−2ε)+2.
Setting t = p(1 − 2)n gives us the following Theorem.
Theorem V.3. Let m be the expected number of edges. If,
m
2
2
≥ (1 + δ)
+
log n,
(21)
n
(1 − 2ε)2
3(1 − 2ε)
then the SDP achieves exact recovery with probability at
least 1 − n−δ . When → 21 , condition (21) is equivalent to
m
(1 + δ)
1
≥2
log n + o
log n.
(22)
n
(1 − 2ε)2
(1 − 2ε)2
Remark V.4. A simpler method to recover xV would be
to fix a vertex-variable, say xVi , to take the value 1. Then
consider, for every vertex j 6= i, all length 2 paths of from
Vertex i to Vertex j and use a voting scheme to determine
xVj . The recovery guarantees we obtained for this method
exhibit the weaker scaling (1 − 2)4 m/n ≥ Ω (log n/n).
ACKNOWLEDGEMENTS
We thank Andrea Montanari for suggesting us the algorithm described in Remark V.4 and Joel Tropp for insightful
discussions regarding [22].
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retrieval with polarization, SIAM J. on Imaging Sci. 7 (2013), no. 1,
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